Strength reduction

In compiler construction, strength reduction is a compiler optimization where expensive operations are replaced with equivalent but less expensive operations.^[1] The classic example of strength reduction converts strong multiplications inside a loop into weaker additions – something that frequently occurs in array addressing.(Cooper, Simpson & Vick 1995, p. 1)

Examples of strength reduction include replacing a multiplication within a loop with an addition and replacing exponentiation within a loop with a multiplication.

Code analysis

Most of a program's execution time is typically spent in a small section of code (called a hot spot), and that code is often inside a loop that is executed over and over.

A compiler uses methods to identify loops and recognize the characteristics of register values within those loops. For strength reduction, the compiler is interested in:

• Loop invariants: the values which do not change within the body of a loop.
• Induction variables: the values which are being iterated each time through the loop.

Loop invariants are essentially constants within a loop, but their value may change outside of the loop. Induction variables are changing by known amounts. The terms are relative to a particular loop. When loops are nested, an induction variable in the outer loop can be a loop invariant in the inner loop.

Strength reduction looks for expressions involving a loop invariant and an induction variable. Some of those expressions can be simplified. For example, the multiplication of loop invariant c and induction variable i

c = 7; for (i = 0; i < N; i++) {  y[i] = c * i; } 

can be replaced with successive weaker additions

c = 7; k = 0; for (i = 0; i < N; i++) {  y[i] = k;  k = k + c; } 

Strength reduction example

Below is an example that will strength-reduce all the loop multiplications that arose from array indexing address calculations.

Imagine a simple loop that sets an array to the identity matrix.

for (i = 0; i < n; i++) {  for (j = 0; j < n; j++)  {  A[i,j] = 0.0;  }  A[i,i] = 1.0; } 

Intermediate code

The compiler will view this code as

0010 ; for (i = 0, i < n; i++) 0020 ; { 0030 r1 = #0  ; i = 0 0040 G0000: 0050 load r2, n ; i < n 0060 cmp r1, r2 0070 bge G0001 0080 0090 ; for (j = 0; j < n; j++) 0100 ; { 0110 r3 = #0  ; j = 0 0120 G0002: 0130 load r4, n ; j < n 0140 cmp r3, r4 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0180 load r7, n 0190 r8 = r1 * r7 ; calculate subscript i * n + j 0200 r9 = r8 + r3 0210 r10 = r9 * #8  ; calculate byte address 0220 fr3 = #0.0 0230 fstore fr3, A[r10] 0240 0250 r3 = r3 + #1  ; j++ 0260 br G0002 0270 ; } 0280 G0003: 0290 ; A[i,i] = 1.0; 0300 load r12, n ; calculate subscript i * n + i 0310 r13 = r1 * r12 0320 r14 = r13 + r1 0330 r15 = r14 * #8  ; calculate byte address 0340 fr4 = #1.0 0350 fstore fr4, A[r15] 0360 0370 ; i++ 0380 r1 = r1 + #1 0390 br G0000 0400 ; } 0410 G0001: 

This expresses 2-dimensional array A as a 1-dimensional array of n*n size, so that whenever the high-level code expresses A[x, y] it will internally be A[(x*n)+y] for any given valid indices x and y.

Many optimizations

The compiler will start doing many optimizations – not just strength reduction. Expressions that are constant (invariant) within a loop will be hoisted out of the loop. Constants can be loaded outside of both loops—such as floating point registers fr3 and fr4. Recognition that some variables don't change allows registers to be merged; n is constant, so r2, r4, r7, r12 can be hoisted and collapsed. The common value i*n is computed in (the hoisted) r8 and r13, so they collapse. The innermost loop (0120-0260) has been reduced from 11 to 7 intermediate instructions. The only multiply that remains in the innermost loop is line 0210's multiply by 8.

0010 ; for (i = 0, i < n; i++) 0020 { 0030 r1 = #0  ; i = 0 0050 load r2, n 0130 ; load r4, n killed; use r2 0180 ; load r7, n killed; use r2 0300 ; load r12, n killed; use r2 0220 fr3 = #0.0 0340 fr4 = #1.0 0040 G0000: 0060 cmp r1, r2 ; i < n 0070 bge G0001 0080 0190 r8 = r1 * r2 ; calculate subscript i * n 0310 ; r13 = r1 * r2 killed; use r8  ; calculate subscript i * n 0090 ; for (j = 0; j < n; j++) 0100 { 0110 r3 = #0  ; j = 0 0120 G0002: 0140 cmp r3, r2 ; j < n 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0200 r9 = r8 + r3 ; calculate subscript i * n + j 0210 r10 = r9 * #8  ; calculate byte address 0230 fstore fr3, A[r10] 0240 0250 r3 = r3 + #1  ; j++ 0260 br G0002 0270 } 0280 G0003: 0290 ; A[i,i] = 1.0; 0320 r14 = r8 + r1 ; calculate subscript i * n + i 0330 r15 = r14 * #8  ; calculate byte address 0350 fstore fr4, A[r15] 0360 0370 ;i++ 0380 r1 = r1 + #1 0390 br G0000 0400 } 0410 G0001: 

There are more optimizations to do. Register r3 is the main variable in the innermost loop (0140-0260); it gets incremented by 1 each time through the loop. Register r8 (which is invariant in the innermost loop) is added to r3. Instead of using r3, the compiler can eliminate r3 and use r9. The loop, instead of being controlled by r3 = 0 to n-1, can be controlled by r9=r8+0 to r8+n-1. That adds four instructions and kills four instructions, but there's one fewer instruction inside the loop.

0110 ; r3 = #0 killed  ; j = 0 0115 r9 = r8 ; new assignment 0117 r20 = r8 + r2 ; new limit 0120 G0002: 0140 ; cmp r3, r2 killed  ; j < n 0145 cmp r9, r20 ; r8 + j < r8 + n = r20 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0200 ; r9 = r8 + r3 killed  ; calculate subscript i * n + j 0210 r10 = r9 * #8  ; calculate byte address 0230 fstore fr3, A[r10] 0240 0250 ; r3 = r3 + #1 killed  ; j++ 0255 r9 = r9 + #1  ; new loop variable 0260 br G0002 

Now r9 is the loop variable, but it interacts with the multiply by 8. Here we get to do some strength reduction. The multiply by 8 can be reduced to some successive additions of 8. Now there are no multiplications inside the loop.

0115 r9 = r8 ; new assignment 0117 r20 = r8 + r2 ; new limit 0118 r10 = r8 * #8  ; initial value of r10 0120 G0002: 0145 cmp r9, r20 ; r8 + j < r8 + n = r20 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0210 ; r10 = r9 * #8 killed  ; calculate byte address 0230 fstore fr3, A[r10] 0240 0245 r10 = r10 + #8  ; strength reduced multiply 0255 r9 = r9 + #1  ; loop variable 0260 br G0002 

Registers r9 and r10 (= 8*r9) aren't both needed; r9 can be eliminated in the loop. The loop is now 5 instructions.

0115 ; r9 = r8 killed 0117 r20 = r8 + r2 ; limit 0118 r10 = r8 * #8  ; initial value of r10 0119 r22 = r20 * #8  ; new limit 0120 G0002: 0145 ; cmp r9, r20 killed  ; r8 + j < r8 + n = r20 0147 cmp r10, r22 ; r10 = 8*(r8 + j) < 8*(r8 + n) = r22 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0230 fstore fr3, A[r10] 0240 0245 r10 = r10 + #8  ; strength reduced multiply 0255 ; r9 = r9 + #1 killed  ; loop variable 0260 br G0002 

Outer loop

Back to the whole picture:

0010 ; for (i = 0, i < n; i++) 0020 { 0030 r1 = #0  ; i = 0 0050 load r2, n 0220 fr3 = #0.0 0340 fr4 = #1.0 0040 G0000: 0060 cmp r1, r2 ; i < n 0070 bge G0001 0080 0190 r8 = r1 * r2 ; calculate subscript i * n 0117 r20 = r8 + r2 ; limit 0118 r10 = r8 * #8  ; initial value of r10 0119 r22 = r20 * #8  ; new limit 0090 ; for (j = 0; j < n; j++) 0100 { 0120 G0002: 0147 cmp r10, r22 ; r10 = 8*(r8 + j) < 8*(r8 + n) = r22 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0230 fstore fr3, A[r10] 0240 0245 r10 = r10 + #8  ; strength reduced multiply 0260 br G0002 0270 } 0280 G0003: 0290 ; A[i,i] = 1.0; 0320 r14 = r8 + r1 ; calculate subscript i * n + i 0330 r15 = r14 * #8  ; calculate byte address 0350 fstore fr4, A[r15] 0360 0370 ;i++ 0380 r1 = r1 + #1 0390 br G0000 0400 } 0410 G0001: 

There are now four multiplications within the outer loop that increments r1. Register r8 = r1*r2 at 0190 can be strength reduced by setting it before entering the loop (0055) and incrementing it by r2 at the bottom of the loop (0385).

The value r8*8 (at 0118) can be strength reduced by initializing it (0056) and adding 8*r2 to it when r8 gets incremented (0386).

Register r20 is being incremented by the invariant/constant r2 each time through the loop at 0117. After being incremented, it is multiplied by 8 to create r22 at 0119. That multiplication can be strength reduced by adding 8*r2 each time through the loop.

0010 ; for (i = 0, i < n; i++) 0020 { 0030 r1 = #0  ; i = 0 0050 load r2, n 0220 fr3 = #0.0 0340 fr4 = #1.0 0055 r8 = r1 * r2 ; set initial value for r8 0056 r40 = r8 * #8  ; initial value for r8 * 8 0057 r30 = r2 * #8  ; increment for r40 0058 r20 = r8 + r2 ; copied from 0117 0058 r22 = r20 * #8  ; initial value of r22 0040 G0000: 0060 cmp r1, r2 ; i < n 0070 bge G0001 0080 0190 ; r8 = r1 * r2 killed  ; calculate subscript i * n 0117 ; r20 = r8 + r2 killed - dead code 0118 r10 = r40 ; strength reduced expression to r40 0119 ; r22 = r20 * #8 killed  ; strength reduced 0090 ; for (j = 0; j < n; j++) 0100 { 0120 G0002: 0147 cmp r10, r22 ; r10 = 8*(r8 + j) < 8*(r8 + n) = r22 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0230 fstore fr3, A[r10] 0240 0245 r10 = r10 + #8  ; strength reduced multiply 0260 br G0002 0270 } 0280 G0003: 0290 ; A[i,i] = 1.0; 0320 r14 = r8 + r1 ; calculate subscript i * n + i 0330 r15 = r14 * #8  ; calculate byte address 0350 fstore fr4, A[r15] 0360 0370 ;i++ 0380 r1 = r1 + #1 0385 r8 = r8 + r2 ; strength reduce r8 = r1 * r2 0386 r40 = r40 + r30 ; strength reduce expression r8 * 8 0388 r22 = r22 + r30 ; strength reduce r22 = r20 * 8 0390 br G0000 0400 } 0410 G0001: 

The last multiply

That leaves the two loops with only one multiplication operation (at 0330) within the outer loop and no multiplications within the inner loop.

0010 ; for (i = 0, i < n; i++) 0020 { 0030 r1 = #0  ; i = 0 0050 load r2, n 0220 fr3 = #0.0 0340 fr4 = #1.0 0055 r8 = r1 * r2 ; set initial value for r8 0056 r40 = r8 * #8  ; initial value for r8 * 8 0057 r30 = r2 * #8  ; increment for r40 0058 r20 = r8 + r2 ; copied from 0117 0058 r22 = r20 * #8  ; initial value of r22 0040 G0000: 0060 cmp r1, r2 ; i < n 0070 bge G0001 0080 0118 r10 = r40 ; strength reduced expression to r40 0090 ; for (j = 0; j < n; j++) 0100 { 0120 G0002: 0147 cmp r10, r22 ; r10 = 8*(r8 + j) < 8*(r8 + n) = r22 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0230 fstore fr3, A[r10] 0240 0245 r10 = r10 + #8  ; strength reduced multiply 0260 br G0002 0270 } 0280 G0003: 0290 ; A[i,i] = 1.0; 0320 r14 = r8 + r1 ; calculate subscript i * n + i 0330 r15 = r14 * #8  ; calculate byte address 0350 fstore fr4, A[r15] 0360 0370 ;i++ 0380 r1 = r1 + #1 0385 r8 = r8 + r2 ; strength reduce r8 = r1 * r2 0386 r40 = r40 + r30 ; strength reduce expression r8 * 8 0388 r22 = r22 + r30 ; strength reduce r22 = r20 * 8 0390 br G0000 0400 } 0410 G0001: 

At line 0320, r14 is the sum of r8 and r1, and r8 and r1 are being incremented in the loop. Register r8 is being bumped by r2 (=n) and r1 is being bumped by 1. Consequently, r14 is being bumped by n+1 each time through the loop. The last loop multiply at 0330 can be strength reduced by adding (r2+1)*8 each time through the loop.

0010 ; for (i = 0, i < n; i++) 0020 { 0030 r1 = #0  ; i = 0 0050 load r2, n 0220 fr3 = #0.0 0340 fr4 = #1.0 0055 r8 = r1 * r2 ; set initial value for r8 0056 r40 = r8 * #8  ; initial value for r8 * 8 0057 r30 = r2 * #8  ; increment for r40 0058 r20 = r8 + r2 ; copied from 0117 0058 r22 = r20 * #8  ; initial value of r22 005A r14 = r8 + r1 ; copied from 0320 005B r15 = r14 * #8  ; initial value of r15 (0330) 005C r49 = r2 + #1 005D r50 = r49 * #8  ; strength reduced increment 0040 G0000: 0060 cmp r1, r2 ; i < n 0070 bge G0001 0080 0118 r10 = r40 ; strength reduced expression to r40 0090 ; for (j = 0; j < n; j++) 0100 { 0120 G0002: 0147 cmp r10, r22 ; r10 = 8*(r8 + j) < 8*(r8 + n) = r22 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0230 fstore fr3, A[r10] 0240 0245 r10 = r10 + #8  ; strength reduced multiply 0260 br G0002 0270 } 0280 G0003: 0290 ; A[i,i] = 1.0; 0320 ; r14 = r8 + r1 killed  ; dead code 0330 ; r15 = r14 * #8 killed  ; strength reduced 0350 fstore fr4, A[r15] 0360 0370 ;i++ 0380 r1 = r1 + #1 0385 r8 = r8 + r2 ; strength reduce r8 = r1 * r2 0386 r40 = r40 + r30 ; strength reduce expression r8 * 8 0388 r22 = r22 + r30 ; strength reduce r22 = r20 * 8 0389 r15 = r15 + r50 ; strength reduce r15 = r14 * 8 0390 br G0000 0400 } 0410 G0001: 

There's still more to go. Constant folding will recognize that r1=0 in the preamble, so several instructions will clean up. Register r8 isn't used in the loop, so it can disappear.

Furthermore, r1 is only being used to control the loop, so r1 can be replaced by a different induction variable such as r40. Where i went 0 <= i < n, register r40 goes 0 <= r40 < 8 * n * n.

0010 ; for (i = 0, i < n; i++) 0020 { 0030 ; r1 = #0  ; i = 0, becomes dead code 0050 load r2, n 0220 fr3 = #0.0 0340 fr4 = #1.0 0055 ; r8 = #0 killed  ; r8 no longer used 0056 r40 = #0  ; initial value for r8 * 8 0057 r30 = r2 * #8  ; increment for r40 0058 ; r20 = r2 killed  ; r8 = 0, becomes dead code 0058 r22 = r2 * #8  ; r20 = r2 005A ; r14 = #0 killed  ; r8 = 0, becomes dead code 005B r15 = #0  ; r14 = 0 005C r49 = r2 + #1 005D r50 = r49 * #8  ; strength reduced increment 005D r60 = r2 * r30 ; new limit for r40 0040 G0000: 0060 ; cmp r1, r2 killed  ; i < n; induction variable replaced 0065 cmp r40, r60 ; i * 8 * n < 8 * n * n 0070 bge G0001 0080 0118 r10 = r40 ; strength reduced expression to r40 0090 ; for (j = 0; j < n; j++) 0100 { 0120 G0002: 0147 cmp r10, r22 ; r10 = 8*(r8 + j) < 8*(r8 + n) = r22 0150 bge G0003 0160 0170 ; A[i,j] = 0.0; 0230 fstore fr3, A[r10] 0240 0245 r10 = r10 + #8  ; strength reduced multiply 0260 br G0002 0270 } 0280 G0003: 0290 ; A[i,i] = 1.0; 0350 fstore fr4, A[r15] 0360 0370 ;i++ 0380 ; r1 = r1 + #1 killed  ; dead code (r40 controls loop) 0385 ; r8 = r8 + r2 killed  ; dead code 0386 r40 = r40 + r30 ; strength reduce expression r8 * 8 0388 r22 = r22 + r30 ; strength reduce r22 = r20 * 8 0389 r15 = r15 + r50 ; strength reduce r15 = r14 * 8 0390 br G0000 0400 } 0410 G0001: 

Other strength reduction operations

This material is disputed. It is better described as peephole optimization and instruction assignment.

Operator strength reduction uses mathematical identities to replace slow math operations with faster operations. The benefits depend on the target CPU and sometimes on the surrounding code (which can affect the availability of other functional units within the CPU).

• replacing integer division or multiplication by a power of 2 with an arithmetic shift or logical shift^[2]
• replacing integer multiplication by a constant with a combination of shifts, adds or subtracts
• replacing integer division by a constant with a multiplication, taking advantage of the limited range of machine integers.^[3] This method also works if divisor is a non-integer sufficiently greater than 1, e.g. √2 or π.^[4]

Original calculation	Replacement calculation
y+= 1	y++
y%2 != 0	y & 1
y = x * 2	y = x << 1
y = x / 2	y = x >> 1
y = x % 2	y = x & 1
y = x * 15	y = (x << 4) - x
y = (uint16_t)x / 10	y = ((uint32_t)x * (uint32_t)0xCCCD) >> 19)
y = (uint16_t)x / π	y = (((uint32_t)x * (uint32_t)0x45F3) >> 16) + x) >> 2)

Induction variable (orphan)

Induction variable or recursive strength reduction replaces a function of some systematically changing variable with a simpler calculation using previous values of the function. In a procedural programming language this would apply to an expression involving a loop variable and in a declarative language it would apply to the argument of a recursive function. For example,

 f x = ... (3 ** x) ... (f (x + 1)) ... 

becomes

 f x = f' x 1  where  f' x z = ... z ... (f' (x + 1) (3 * z)) ... 

Here modified recursive function f′ takes a second parameter z = 3 ** x, allowing the expensive computation (3 ** x) to be replaced by the cheaper (3 * z).

See also

• Fast inverse square root
• Hacker's Delight
• Partial evaluation

Notes

1. Steven Muchnick; Muchnick and Associates (15 August 1997). Advanced Compiler Design Implementation. Morgan Kaufmann. ISBN 978-1-55860-320-2. Strength reduction.
2. In languages such as C and Java, integer division has round-towards-zero semantics, whereas a bit-shift always rounds down, requiring special treatment for negative numbers. For example, in Java, -3 / 2 evaluates to -1, whereas -3 >> 1 evaluates to -2. So in this case, the compiler cannot optimize division by two by replacing it by a bit shift.
3. Granlund, Torbjörn; Peter L. Montgomery. "Division by Invariant Integers Using Multiplication" (PDF).
4. Jones, Nigel. "Division of integers by constants".

References

• Aho, Alfred V.; Sethi, Ravi; Ullman, Jeffrey D. (1986), Compilers: Principles, Techniques, and Tools (2nd ed.), ISBN 978-0-201-10088-4
• Allen, Francis E.; Cocke, John; Kennedy, Ken (1981), "Reduction of Operator Strength", in Munchnik, Steven S.; Jones, Neil D. (eds.), Program Flow Analysis: Theory and Applications, Prentice-Hall, ISBN 978-0-13-729681-1
• Cocke, John; Kennedy, Ken (November 1977), "An algorithm for reduction of operator strength", Communications of the ACM, 20 (11): 850–856, doi:10.1145/359863.359888, S2CID 1092505
• Cooper, Keith; Simpson, Taylor; Vick, Christopher (October 1995), Operator Strength Reduction (PDF), Rice University, retrieved April 22, 2010